Mode matched y-junction

ABSTRACT

A mode-matched waveguide Y-junction with balanced or unbalanced splitting comprises an input waveguide, expanding from an input end to an output end, for expanding the input beam of light along a longitudinal axis; first and second output waveguides extending from the output end of the input waveguide separated by a gap. Ideally, each of the first and second output waveguides includes an initial section capable of supporting a fundamental super mode, and having an inner wall substantially parallel to the longitudinal axis, and a mode splitting section extending from the initial section at an acute angle to the longitudinal axis.

This application is a continuation of U.S. patent application Ser. No. 15/928,594, now allowed, which is a continuation of and claims priority to and the benefit of U.S. patent application Ser. No. 15/423,843, now U.S. Pat. No. 9,946,020, which applications are incorporated herein by reference in their entireties.

TECHNICAL FIELD

The present invention relates to a waveguide Y-junction, and in particular to a mode-matched waveguide Y-junction with balanced or unbalanced splitting.

BACKGROUND

With reference to FIGS. 1a, 1b and 1c , a junction section (JS) 4 of a conventional semiconductor waveguide Y-junction splitter 1 receives an input beam at an input port 2 for transmission to an input waveguide 3, and splits the input beam into two output beams onto two output waveguides 6 and 7 for output to two output ports 8 and 9, respectively. Typically, the input port 2 and the output ports 8 and 9 are optically coupled to external waveguides (not shown) for transmitting the optical beams to and from the Y-junction splitter 1. A good Y-junction splitter is characterized by low insertion loss (IL), i.e. the amount of power lost through the Y-junction splitter that does not go to the output waveguides 8 and 9; low return loss (RL), the amount of light reflected by the JS 4; and good split ratio, e.g. a balanced Y-junction splits evenly 50:50, not 51:49.

One problem that arises, especially with a high-index contrast platform, such as Si/SiO2 or SiN/SiO2, is that there can be an abrupt change in mode profile between the optical mode guided just before the JS 4 and the optical mode just after the JS 4. The abrupt change results in exciting multiple modes past the JS 4, such as high order guided modes or radiation modes. These parasitic modes can lead to high IL or RL.

Another problem arises with the design of unbalanced Y-junctions splitters. Although balanced Y-junction splitters are intuitively designed by symmetry, designing an unbalanced Y-junction splitter with an arbitrary split ratio is non-trivial; especially when low IL is required.

An object of the present invention is to overcome the shortcomings of the prior art by providing a more efficient Y-junction waveguide splitter.

SUMMARY OF THE INVENTION

Accordingly the present invention relates to a splitter for splitting an input beam of light into first and second portions comprising: an input port for launching the input beam of light; an input waveguide including a longitudinal axis, capable of providing adiabatic expansion of the input beam of light from an input end proximate the input port to an output end, which has a width that supports a fundamental mode and a second order mode; first and second output waveguides extending from the output end of the input waveguide on either side of the longitudinal axis, and separated by a gap; and first and second output ports for outputting the first and second portions, respectively. Each of the first and second output waveguides includes: an initial section, including an initial width smaller than the input end of the input waveguide, whereby the initial sections and the gap form a single waveguide capable of supporting a super mode of the input beam of light, which spans the initial sections of the first and second output waveguides and the gap; a mode splitting section extending from the initial section at an acute angle to the longitudinal axis for splitting the super mode of the input beam of light into first and second portions; and a final expansion section extending from the mode splitting section, at least one of the final expansion sections including an expanding width expanding to a same width as the input end of the input waveguide.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described in greater detail with reference to the accompanying drawings which represent preferred embodiments thereof, wherein:

FIG. 1a is a schematic plan view of a junction split section of a conventional Y-junction splitter.

FIG. 1b is a cross-sectional view of the input waveguide of the Y-junction splitter of FIG. 1a illustrating light intensity.

FIG. 1c is a cross-sectional view of the input waveguide of the Y-junction splitter of FIG. 1a at the junction split illustrating light intensity.

FIG. 2a is a schematic plan view of a junction split section of a Y-junction splitter in accordance with the present invention.

FIG. 2b is a cross-sectional view of the input waveguide of the Y-junction splitter of FIG. 2a illustrating light intensity.

FIG. 2c is a cross-sectional view of the input waveguide of the Y-junction splitter of FIG. 2a at the junction split section illustrating light intensity.

FIG. 3 is a schematic plan view of a Y-junction splitter in accordance with the present invention.

FIG. 4 is a schematic plan view of a Y-junction splitter in accordance with another embodiment of the present invention.

FIG. 5 is a cross-sectional view of the input waveguide of the Y-junction splitter of FIG. 4 at the junction split illustrating light intensity.

DETAILED DESCRIPTION

While the present teachings are described in conjunction with various embodiments and examples, it is not intended that the present teachings be limited to such embodiments. On the contrary, the present teachings encompass various alternatives and equivalents, as will be appreciated by those of skill in the art.

With reference to FIGS. 2a, 2b, 2c and 3, a mode-matched semiconductor waveguide Y-junction splitter 11 includes a junction split section 14, which receives an input beam at an input port 12 for transmission to an input waveguide 13, and splits the input beam into two output beams onto two output waveguides 16 and 17 for output to two output ports 18 and 19, respectively. The input port 12 may be along a longitudinal axis 15 of the Y junction splitter 11, which may divide the input waveguide 13 into two symmetrical halves, and provide an axis of symmetry for the two output waveguides 16 and 17.

Ideally, the Y-junction splitter 11 may be comprised of a silicon on insulator (SOI) waveguide, including a high index silicon core sandwiched between upper and lower cladding layers, comprised of lower index silica; however, other forms and semiconductor materials, e.g., high index cores with refractive indexes between 3 and 4, such as SiN or other Group III/V materials, sandwiched between lower index upper and lower cladding layers with refractive indexes between 1 and 2, such as nitride or other Group III/V materials, are within the scope of the invention. Typically, the input port 12 and the output ports 18 and 19 are optically coupled to external waveguides (not shown) for transmitting the optical beams to and from the Yjunction 11.

The input waveguide 13 expands the input beam of light adiabatically from an input end to an output end, whereby the width of the output end of the input waveguide 13 supports two guided modes (see FIG. 2b ), e.g., the fundamental mode TE0 (or TM0 depending on polarization) and the second order mode TE1 (or TM1 depending on polarization). There is no need to expand the input waveguide 13 to a width that would support more than two guided modes unless one seeks splitting in more than two output waveguides 16 and 17. The expansion is smooth and long enough, such that light remains in the TE0 mode (a.k.a. adiabatic taper) despite the presence of the second order mode.

At the JS 14, a super or hybrid mode is supported, which is defined by an arrangement of waveguides that are in close proximity, such that they share the same optical mode. Ideally, the geometry of the super mode, to the right of the JS 14, is optimized to achieve maximal overlap integral with the incoming mode of the input waveguide 13 just left to the JS 14 in order to define a mode-matched junction. (see FIG. 2c ) A gap 21 between the first and second output waveguides 16 and 17 on the right-hand side of the JS 14 is typically chosen to be as small as the fabrication process allows, e.g. less than 500 nm, ideally between 50 nm and 300 nm. The gap 21 may be larger if coupling from another mode, other than the fundamental mode, is desirable. An important aspect is that the geometry of the super mode should be selected such that it maximizes the overlap integral between the incoming mode (left of JS 14) and the super mode (right of the JS 14). With reference to FIG. 2c , the illustrated super mode extends across the gap 21 and into both the first and second output waveguides 16 and 17 on either side thereof, in contrast to FIG. 1c , which illustrates two distinct modes in the conventional Y-junction splitter.

Accordingly, at the JS 14 the incoming mode, e.g. TE0, is on the left side and the goal is to engineer the dimensions, e.g. widths of the gap 21 and the initial sections of the output waveguides 16 and 17, such that the incoming TE0 mode excites, as much as possible, the new TE0 super mode supported by the input waveguides 16 and 17. Accordingly, the overall insertion loss of the splitter 11 is kept low by selecting a geometry at the JS 14, such that almost all of the incoming light, e.g., TE0 mode, propagates to the super TE0 mode. This is achieved by first selecting a width of the gap 21 that is as small as possible, e.g., less than 500 nm, ideally between 50 nm and 300 nm, in order to have the super mode fully supported by the initial sections of the input waveguides 16 and 17. Then the widths of at least the initial sections, e.g., up to the entire length, of the output waveguides 16 and 17, are varied while tracking the overlap integral between the incoming TE0 mode at the JS 14 and the super TE0 after the JS 14. Mode matching is achieved when the overlap between these two modes is maximized, resulting in a discontinuity between the input waveguide 13 and the initial sections of the output waveguides 16 and 17 that can look counter-intuitive.

Therefore, past the JS 14 in the initial sections of the output waveguides 16 and 17 most of the light is still confined in the super TE0 mode. When it is in this state it is easy to move the spatial power distribution of the mode by changing the size of the output waveguides 16 and 17. The gap 21 is not increased at least in the initial section in order to maintain the hybridization of the modes, i.e., the initial sections of the output waveguides 16 and 17 are close enough together such that, from the optical stand point, they form a single waveguide.

The mode-matching structure of the JS 14 between the input waveguide 13, and the first and second output waveguides 16 and 17 is provided in order to achieve mode matching between the expanded single mode beam at the output of the input waveguide 13, and the super mode at the input of the first and second output waveguides 16 and 17.

Typically, the outer edge of a Y-junction is a smooth curve; however, in the illustrated example, the width of the input waveguide 13 is wider than the combined widths of the first and second output waveguides 16 and 17 and the gap 21, resulting in a discontinuity in the form of stepped shoulders at the JS 14 on opposite sides thereof adjacent the first and second output waveguides 16 and 17, each shoulder including a surface perpendicular to the direction of light propagation. An added advantage of a junction geometry that accomplishes mode matching is the resilience to fabrication processes that its stable maxima brings (e.g., deviation from the optimal geometry will result in added loss due to mode mismatch, but the added loss will be small around the maxima point.) In an exemplary embodiment, the input waveguide 13 may be 2700 nm wide, while the first and second output waveguides 16 and 17 may be 675 nm wide with a 300 nm gap therebetween, leaving a 525 nm shoulder, e.g., a surface perpendicular to the longitudinal axis 15 adjacent outer sides of the output waveguides 16 and 17, providing the discontinuity. The shoulder may be wider than the gap 21, but not as wide as the output waveguides 16 and 17. The width at the output end of the input waveguide 13 is about twice as wide as the combined width of the first and second output waveguides 16 and 17.

In an alternative embodiment, the mode-matching JS 14 includes a subwavelength gratings in the gap 21 alternative to or in combination with the aforementioned discontinuity. Utilizing a high resolution process, such as electron beam lithography, the subwavelength structures (metamaterial) can be provided to have a very fine grating (˜10× smaller than the effective wavelength guided in the input waveguide 13) This effectively increases the effective index of the gap 21, which helps achieve mode matching to the incoming waveguide 13. Such a variation would reduce the height of the shoulder width, which could increase the widths of the output waveguides to approach the full width of the input waveguide 13. The subwavelength gratings could start with a high fill ratio in the gap 21 (more high index material than low in a given period of the grating) and adiabatically taper to a low fill ratio (more low index material than high) at the limit of which a full gap would open between the two output waveguides 16, 17.

Another variation includes splitting the input beam into a plurality, e.g., more than two, output waveguides; whereby the same mode matching technique may be employed at the JS 14.

With reference to FIG. 3, the Y-junction splitter 11 of the present invention comprises the input port 12 optically coupled to an input waveguide 13, which gradually expands from an input end proximate the input port 12 to the junction split section 14 at an output end. The input waveguide 13 expands linearly or exponentially with a gradually decreasing slope. However, any other smooth taper shape may be used provided that it is long enough to allow adiabatic mode expansion. The expansion is by at least a factor of 2 or between 1.5 and 2.5, e.g., from a 1.2 μm input end to a 2.7 μm output end. An input beam of light is launched via the input port 12 from an external source, and propagates into the input waveguide 13 to the junction split (JS) section 14 along a longitudinal axis 15 and in the direction of propagation, and expands adiabatically to the output end of the input waveguide 13. The input port 12 may be along the longitudinal axis 15 of the Y junction splitter 11, which may divide the input waveguide 13 into two symmetrical halves, and may provide an axis of symmetry for the entire Y-junction splitter 11.

The width of the input waveguide port 12 is preferably narrow enough to provide propagation of only a single mode of the input beam. The width of the input waveguide 13 expands providing adiabatic expansion in the core region to a width that supports two guided modes, e.g., the fundamental mode TE0 (or TM0 depending on polarization) and the second order mode TE1 (or TM1 depending on polarization). There is no need to expand the input waveguide 13 to a width that would support more than two guided modes unless one seeks splitting in more than two output waveguides 16 and 17. The expansion is smooth and long enough, such that light remains in the TE0 mode (a.k.a. adiabatic taper) despite the presence of the second order mode.

First and second output waveguides 16 and 17, respectively, extend from the output end of the input waveguide 13 with a gap 21 therebetween, symmetrical about the longitudinal axis 15, for receiving first and second portions of the input beam according to a desired splitting ratio, e.g., 50/50, and for outputting the first and second portions to output ports 18 and 19, respectively.

As above, the structure of the mode-matching JS 14 provides that the width of the input waveguide 13 is wider than the combined widths of the first and second output waveguides 16 and 17 and the gap 21, resulting in a mode-matching discontinuity, e.g., a stepped shoulders at the JS 14 on opposite sides thereof adjacent the first and second output waveguides 16 and 17, each shoulder including a surface perpendicular to the direction of light propagation 15. The shoulder may be wider than the gap 21, but may not as wide as the output waveguides 16 and 17. The width at the output end of the input waveguide 13 is about twice as wide as the combined width of the first and second output waveguides 16 and 17.

Each of the first and second output waveguides 16 and 17 are divided into plurality of sections 36 a to 36 d and 37 a to 37 d, respectively. Initial sections 36 a and 37 a may be straight sections parallel to the longitudinal axis 15 with a constant gap width, e.g., between 50 nm to 500 nm, but ideally less than 300 nm, therebetween, and each preferably having a constant width, e.g., 100 nm to 300 nm for Si/SiO2 or 500 nm to 800 nm for SiN/SiO2, enabling the input mode to stably transform into a super mode. The length of the initial sections 36 a and 37 a are typically less than or equal to one half the length of the input waveguide region 13 or in absolute terms preferably greater than 1 μm, preferably less than 20 μm, and preferably between 5 μm and 10 μm.

Accordingly, at the JS 14 the incoming mode, e.g., TE0, is on the left side and the goal is now to engineer the dimensions, e.g., gap and waveguide widths, of the initial sections of the output waveguides 36 a and 37 a, such that the incoming TE0 mode excites, as much as possible, the new TE0 super mode supported by the initial sections 36 a and 37 a. Accordingly, the overall insertion loss of the Y-junction splitter 11 is kept low by selecting a geometry at the JS 14, such that almost all of the incoming light, e.g., TE0 mode, propagates to the super TE0 mode. This is achieved by first selecting a width of the gap 21 that is as small as possible, e.g., 50 nm to 300 nm, in order to have the super mode fully supported by the initial sections 36 a and 37 a. Then the widths of the initial sections 36 a and 37 a are varied while tracking the overlap integral between the incoming TE0 mode and the super TE0 mode. Mode matching is achieved when the overlap between these two modes is maximized, resulting in a discontinuity between the input waveguide 13 and the initial sections 36 a and 37 a of the output waveguides 16 and 17, respectively, that can look counter-intuitive.

Therefore, past the JS 14 in the initial sections 36 a and 37 a most of the light is still confined in the super TE0 mode. When it is in this state it is easy to move the spatial power distribution of the mode by tapering the size of the initial sections 36 a and 37 a. The gap 21 is not increased in the initial sections 36 a and 37 a in order to maintain the hybridization of the modes, i.e., the initial sections 36 a and 37 a of the output waveguides 16 and 17 are close enough together such that, from the optical stand point, they form a single waveguide.

Once the light is in the super mode, past the JS 14, a V-splitter is used to split the mode into the two mode splitting sections 36 b and 37 b. The best way of achieving this is using a V-shaped split, in contrast to a typical s-bend. The V-splitter shape constantly separates the two mode splitting sections 36 b and 37 b at an acute angle, e.g., 3° to 15°, from the longitudinal axis 35, and linearly decouples the output waveguides 36 and 37, which is found to be simpler and more efficient than an otherwise S-bend. Furthermore, the S-bend may result in high order mode coupling, if the bend radius is not large enough; which is avoided using a V-splitter.

Final expansion sections 36 c and 37 c gradually, e.g., linearly, expand the first and second output waveguides 16 and 17, respectively, by a factor of approximately 1.3 to 2, e.g., from 500 nm to 800 nm to 700 μm to 1.6 μm. Ideally, the inner walls of the final expansion sections 36 c and 37 c extend coplanar, and are separated by the same angle as the mode splitting sections 36 b and 37 b; however, any structure is within the scope of the invention. Final expansion sections 36 c and 37 c are used in the V-splitter, once the gap between the input and output waveguide arms 16 and 17 is sufficiently large, such that it avoids having mode overlap.

Bend sections 36 d and 37 d redirect the ends of the first and second output waveguides 16 and 17, respectively, to be back parallel with the longitudinal axis 15, and at the same width as the end of the taper sections 36 c and 37 c. At the output ports 18 and 19 of the Y-junction 11, the output waveguides 16 and 17 are back to the initial width, e.g., for single mode propagation, so that the input and output ports 12, 18 and 19 have the same widths, e.g., the best waveguide width for low loss propagation.

With reference to FIGS. 4 and 5, another embodiment of the Y-junction splitter 41 of the present invention comprises an input port 42 optically coupled to an input waveguide 43, which gradually expands from an input end proximate the input port 42 to a junction split section 44 at an output end. The input waveguide 43 expands linearly or exponentially with a gradually decreasing slope. The expansion is by at least a factor of 2 or between 1.5 and 2.5, e.g., from a 1.2 μm input end to a 2.7 μm output end. An input beam of light is launched via the input port 42 from an external source, and propagates into the input waveguide 43 to the junction split (JS) section 44 along a longitudinal axis 45 and in the direction of propagation, and expands adiabatically to the output end of the input waveguide 43. First and second output waveguides 46 and 47, respectively, extend from the output end of the input waveguide 43 for receiving first and second portions of the input beam according to a desired unbalanced splitting ratio, e.g., 25/75, and for outputting the first and second portions to output ports 49 a and 49 b, respectively.

As above, with reference to FIGS. 2 and 3, the structure of the mode-matching JS 41 provides that the width of the input waveguide 43 is wider than the combined widths of the first and second output waveguides 46 and 47 and the gap 48, resulting in a mode-matching discontinuity, e.g., stepped shoulders at the JS 44 on opposite sides thereof adjacent the first and second output waveguides 46 and 47, each shoulder including a surface perpendicular to the direction of light propagation 45. The shoulder may be wider than the gap, but not as wide as the output waveguides 46 and 47. The width at the output end of the input waveguide 43 is about twice as wide as the combined width of the first and second output waveguides 46 and 47.

In an alternative embodiment, the mode-matching JS 44 may include a subwavelength gratings in the gap 48 alternative to or in combination with the aforementioned discontinuity, as hereinbefore disclosed with reference to the JS 14.

Each of the first and second output waveguides 46 and 47 are divided into a plurality of sections 46 a to 46 d and 47 a to 47 d, respectively. Initial sections 46 a and 47 a form an unbalanced region, as section 46 a comprises a straight sections parallel to the longitudinal axis 45 with a constant width, e.g., 100 nm to 300 nm for Si/SiO2 or 500 nm to 800 nm for SiN/SiO2, while section 47 a comprises an initial expansion section, linearly expanding from the JS 44 to section 47 b. Ideally, the inner walls of the initial unbalanced sections 46 a and 47 a are separated by a constant gap width, e.g., between 50 nm to 500 nm, but ideally less than 300 nm, while the outer wall of the initial section 47 b tapers away from the longitudinal axis 45; however, other arrangements are within the scope of the invention. The length of the initial sections 46 a and 47 a are typically between one half to two times the length of the Y-junction core region 43 or in absolute terms preferably at least 5 μm, preferably less than 20 μm, and preferably between 5 μm and 10 μm.

In order to create the unbalanced Y-junction 41, an unbalanced region is used where light is confined in a hybrid mode or super mode. The unbalanced region is defined as a region where the width of one of the junction arms, i.e., 47 a, is tapered, i.e., expands. This breaks the symmetry of the super mode adiabatically, which results in shifting the power distribution towards one arm or another (See FIG. 5). Arbitrary splitting ratios can be achieved by properly selecting the un-balanced region's length and by tapering the symmetrical super mode to an asymmetrical with sought power distribution.

Once the light is in the super mode, past the JS 44, a V-splitter is used to split the mode into the two mode splitting sections 46 b and 47 b. The V-splitter shape constantly separates the two mode splitting sections 46 b and 47 b at an acute angle, e.g., 3° to 15°, from the longitudinal axis 45, and linearly decouples the output waveguides 46 and 47, which is found to be simpler and more efficient than an otherwise S-bend. Furthermore, the S-bend may result in high order mode coupling, if the bend radius is not large enough; which is avoided using a V-splitter. The V-splitter sections 46 b and 47 b have constant, but different widths, e.g., 46 b is 500 nm to 800 nm wide, while 47 b is 700 μm to 1.6 μm wide.

Final expansion section 46 c gradually, e.g., linearly, expands the first output waveguides 36 by a factor of approximately 1.3 to 2, e.g., from 500 nm to 800 nm to 700 μm to 1.6 μm, to equal the expansion provided by the initial expansion section 47 a. Ideally, the inner walls of the final expansion sections 46 c and 47 c extend coplanar, and are separated by the same angle as the mode splitting sections 46 b and 47 b; however, any structure is within the scope of the invention. Final expansion sections 46 c and 47 c are used in the V-splitter, once the gap between the input and output waveguide arms 46 and 47 is sufficiently large, such that it avoids having mode overlap.

Bend sections 46 d and 47 d redirect the ends of the first and second output waveguides 46 and 47, respectively, to be back parallel with the longitudinal axis 45, and at the same width as the end of the final expansion sections 46 c and 47 c.

Lithographic rounding at the interfaces may be included to provide a smooth transition between sections.

The foregoing description of one or more embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto. 

1-20. (canceled)
 21. A splitter for splitting an input beam of light into first and second portions comprising: an input port for launching the input beam of light; an input waveguide including a longitudinal axis, capable of providing adiabatic expansion of the input beam of light from an input end proximate the input port to an output end, which has a width that supports a fundamental mode and a second order mode; first and second output waveguides extending from the output end of the input waveguide on either side of the longitudinal axis, and separated by a gap; and first and second output ports for outputting the first and second portions, respectively; wherein each of the first and second output waveguides includes: an initial section, including an initial width smaller than the input end of the input waveguide, whereby the initial sections and the gap form a single waveguide capable of supporting a super mode of the input beam of light, which spans the initial sections of the first and second output waveguides and the gap; a mode splitting section extending from the initial section at an acute angle to the longitudinal axis for splitting the super mode of the input beam of light into first and second portions; and a final expansion section extending from the mode splitting section, at least one of the final expansion sections including an expanding width expanding to a same width as the input end of the input waveguide.
 22. The splitter according to claim 21, wherein the gap includes a constant width between 50 nm to 500 nm wide.
 23. The splitter according to claim 21, wherein the gap is less than 300 nm wide.
 24. The splitter according to claim 21, wherein each final expansion section expands by a factor of 1.3 to 2 to the same width as the input end of the input waveguide.
 25. The splitter according to claim 24, wherein each final expansion section expands from between 500 nm and 800 nm to between 700 nm and 1.6 μm.
 26. The splitter according to claim 21, wherein the initial width of the initial section of each of the first and second output waveguides is smaller than the width of the input end of the input waveguide by a factor or 1.3 to
 2. 27. The splitter according to claim 21, wherein the initial section of each of the first and second output waveguides includes a constant width of between 100 nm and 300 nm.
 28. The splitter according to claim 21, wherein the initial section of each of the first and second output waveguides includes a constant width of between 500 nm and 800 nm.
 29. The splitter according to claim 21, wherein the initial section of the first output waveguide comprises a constant width; and wherein the initial section of the second output waveguide comprises an expanding width to create an un-balanced power distribution at the output waveguides.
 30. The splitter according to claim 29, wherein the initial section of the second output waveguide expands by a factor of 1.3 to
 2. 31. The splitter according to claim 30, wherein the final expansion section of the first output waveguide expands by a factor of 1.3 to
 2. 32. The splitter according to claim 21, wherein the gap includes a subwavelength grating for mode matching the input waveguide and the first and second output waveguides.
 33. The splitter according to claim 21, wherein the acute angle is between 3° and 15°.
 34. The splitter according to claim 21, wherein the output end of the input waveguide is wider than a combined width of the initial widths of the initial sections and the gap resulting in a discontinuity in the input waveguide to provide mode matching between the input waveguide and the first and second output waveguides.
 35. The splitter according to claim 34, wherein the output end of the input waveguide is about twice as wide as a combined width of input ends of the first and second output waveguides.
 36. The splitter according to claim 21, wherein the input waveguide expands by at least a factor of
 2. 37. The splitter according to claim 21, wherein the input waveguide expands by a factor of 1.5 to 2.5.
 38. The splitter according to claim 21, wherein the input waveguide expands from about 1.2 μm to about 2.7 μm.
 39. The splitter according to claim 21, wherein the initial sections are less than one half a length of the input waveguide.
 40. The splitter according to claim 21, wherein the initial sections are between 5 μm and 10 μm in length. 